We study compact Kähler manifolds X admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of {tangential \ deformations} , and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of X . We extend Calabi's theorem on the structure of compact Kähler manifolds X with c_1(X) =0 to compact Kähler manifolds with nonvanishing tangent fields, proving that any such manifold X admits an arbitrarily small tangential deformation which is a suspension over a torus; that is, a quotient of F\times \mathbb C^s fibering over a torus T=\mathbb C^s/\Lambda . We further show that either X is uniruled or, up to a finite Abelian covering, it is a small deformation of a product F\times T where F is a Kähler manifold without tangent vector fields and T is a torus. A complete classification when X is a projective manifold, in which case the deformations may be omitted, or when \dim X\leq s+2 is also given. As an application, it is shown that the study of the dynamics of holomorphic tangent fields on compact Kähler manifolds reduces to the case of rational varieties.